Consider a disk of radius $r$ centered at $(x,y)$, where $(x,y)$ is chosen from the uniform distribution on $[0,1) \times [0,1)$, and let the random variable $N$ be the number of lattice points in the ...

Title question description: Select two lattices $\Lambda_1$ and $\Lambda_2$ (here a lattice=additive free abelian group without accumulation points) of maximal rank two in the real plane. We normalize ...

In answer to Pete L. Clark's question Must a ring which admits a Euclidean quadratic form be Euclidean? on Euclidean quadratic forms, I also gave an example in six or fewer variables, repeated below. ...

What is the highest dimension for which the space of reduced positive definite quadratic forms (or the fundamental domain of $SL_n(\mathbb{R})/SL_n(\mathbb{Z})$) has been explicitly calculated? I know ...

There are arbitrarily large finite sets of points in $\mathbb R^3$ whose Voronoi-domains
intersect all pairwise in $2-$dimensional polytopes. This shows that one needs infinitely many
colours in order ...

Given an operator $f\colon R^m\to R^n$, can one always find a non-zero vector
$x\in \{ 0,1 \}^m$ such that $\|f(x)\|/\|x\|\ge0.01\|f\|$? (Here I denote by
$\|\cdot\|$ both the Euclidean norms in $R^m$ ...

Suppose I want to construct an $N$-gon in the plane whose vertices are integer lattice points, and which is close to a regular $N$-gon (which means, the ratio of longest to the shortest side is within ...

I have some (rather exotic) subspace $L<R^n$, and I want to show that every non-zero vector in $\{0,1\}^n$ has a relatively small projection onto $L$. What general results and tools can be helpful? ...

Two convex lattice polygons are equivalent if there is a lattice-preserving affine transformation mapping one of them to the other. Equivalent polygons have the same area. Let $H(A)$ denote the number ...

I recently became engaged in the work of Siegel, Schmidt, Rogers, Macbeath regarding random lattices and geometry of numbers, e.g. Siegel proved that
$$\int_{SL(n,\mathbb{R})/SL(n,\mathbb{Z})} \sum_{ ...

Hello everyone, I was doing some late night random reading and I got to wonder about some stuff about the Gauss circle problem.
To begin with, consider a circle in $\mathbb{R}^{2}$ with centre at the ...

Given a (finite-dimensional) lattice $L$ of an Euclidean vector-space, the function
$$L\longmapsto -\log(\hbox{packing density of }L)/
\log(\hbox{covering density of }L)$$
is bounded and bounded away ...